Portfolio Management - Lecture notes - ln1 - 6 PDF

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Lecture 1 Introduction and Bond Pricing

Lecturers • The first part of the course is taught by Joakim Bang – Office hours: Mondays 3 pm – 5 pm, room 311, ASB

• The second part is taught by Sidharth Sahgal – Office hours TBA on blackboard

Tutorials • Starts Monday next week • Attendance and preparation is mandatory – Tutorial participation counts for 5 % of your final grade

• Exercises are from the textbook and available in the course outline (or TBA for later tutorials)

Textbook • We use Bodie, Kane and Marcus Investments, 9th edition • You may or may not get by with an earlier edition of the book • The custom textbook “FINS2624” is just the prescribed chapters from the full book • There is no reason to get the full book for this course, but you may need it for later courses

Online Quizzes • Every week, starting week 1, there will be an online quiz that counts towards your final grade • You get a maximum of three attempts, with the highest score counting • There’s a total of 11 quizzes, counting for a total of 11 % of your final grade • The quizzes will open Friday night, and you will be given one week to solve it • Everybody gets an automatic extension of one week. Further extensions will only be given with a doctor’s certificate for both weeks.

Group assignment • The assignment is due via blackboard 9 a.m. on Monday September 10 (week 8) • Please form groups with one to three members within your tutorial group • The assignment will be available on blackboard in due time and the solution is discussed in the week 8 tutorials • The assignment counts for 10 % of the final grade

Midterm and finals • The midterm counts for 37 % of the final grade and is given in class week 8 – It covers material from the lectures in weeks 1 to 6

• The final exam counts for 37 % and is given in the final exam period – It focuses heavily on the lecture material not tested in the midterm, but you are expected to master everything covered in the lectures, tutorials and assignment

First topic: Bond pricing • We will approach this topic a bit differently than the textbook • Specifically, we will be explicit about how arbitrage pricing allows use discounting to price a bond • The lecture notes will give you a good understanding about what is relevant for the midterm • Some of the extensions in the textbook will be covered in the tutorial sessions

What is a bond? • A claim on some fixed future cash flow(s), CF. • The bond matures at the time of its last cash flow, T. • Typically a “large” cash flow at maturity. We call this the par value or face value (FV). • There may be a series of smaller cash flows before maturity. We call these coupons. • There may be zero, one or more coupons in a given year.

What is a bond? • The sum of the annual coupons are often expressed as a fraction of the FV, e.g. 5 %. We call this the coupon rate (C). Let’s denote the actual coupon, e.g. $5, with ct, where t is the period in which we get the coupon. • A bond with no coupons is called a zerocoupon bond

Cash flows of a bond • This figure illustrates the cash flows of a bond with a FV of 100 and a yearly coupon of 5 -P

c1

c2 FV

-91.3

5

5 100

Default risk • That somebody promises to pay you some money doesn’t necessarily mean they will • The risk that you will be unable to collect your cash flows is called default risk • This is very important in practice, but we will generally ignore it in this course

Other frequent assumptions • • • •

No transaction costs Constant interest rates Complete markets These are all true within our model – Compare this to the assumption of vacuum in classical mechanics

Two approaches to pricing • Fundamental pricing – Prices are set in a supply-demand equilibrium – The properties of an asset tell us what that price is likely to be – We will use this approach when pricing stocks later in the course

• Arbitrage pricing – Take some price as given and price other assets relative to that – We will use this approach when pricing bonds and derivatives

What is arbitrage? • An arbitrage is a (set of) trades that generate zero cash flows in the future, but a positive and risk free cash flow today • This is the proverbial “free lunch” or “money machine” • A simple example exploits violations of the law of one price, e.g. an identical bond selling for two different prices – Simultaneously buying the cheap bond and selling the expensive bond would be an arbitrage trade

• All arbitrage pricing is priced based on the same principle, but the trades are (slightly) more complex

Replicating portfolios • We typically rely on a portfolio of assets that exactly mimic the cash flows of some other asset • We call such portfolios replicating portfolios or synthetic assets • Arbitrage pricing is all about constructing replicating portfolios using assets with known prices

Example: Pricing a zero-coupon bond • How would you price the risk-free one-year zero-coupon bond below? Bond A 100

Example: Pricing a zero-coupon bond • You may already know how to discount the future cash-flow with some appropriate discount rate, r, to get the present value • Assuming that r = 10% you’d get PA =

100 100 = ≈ 90.9 + r 1 1.10 ( )

• What is the economic logic behind this?

Where does the discount rate come from? • The appropriate discount rate, r, is the return we could have earned at some alternative investment with the same risk • Let’s say there’s a bank where you can lend and borrow money at 10% interest • Suppose the price of Bond A was actually $80.9, i.e. lower than what we found on the last slide

Constructing a replicating portfolio • We know that the bond is mispriced. How do we exploit this? • We want to make a synthetic version of the bond, i.e. some investments that mimic its cash flows exactly • In this simple example we can just put some amount of money, M, in the bank. • How large must M be? • After one year in the bank account earning 10% interest, it should have grown to match the bonds cash flow of $100 • We must have 1.1M = 100 100 ≈ 90.9 M= 1.1

Exploiting the mispricing • The $90.9 bank deposit replicates the bonds cash flow (is a synthetic bond) but has a different price • We buy the cheap instrument and sell the expensive (in this case the synthetic) instrument • “Selling” a bank deposit means borrowing the money

What are our cash flows? • Today we borrow $90.9 and buy the bond for $80.9. We are left with $10. • In one year the bond pays us $100 which is exactly enough to repay the loan. We have zero net cash flow. • Our “free” $10 is an arbitrage profit and the entire scheme is an arbitrage trade

Arbitrage pricing • In practice smart people will identify arbitrage opportunities and trade on them • This will increase the demand for the bond and raise its price until no further arbitrage trades are possible, i.e. until prices are in equilibrium • In this course we are interested in finding those equilibria, e.g. arbitrage-free prices • We can not say whether it was the bond price or the bank’s interest rate that was wrong • We can only say (and only care) if the prices are internally consistent

How do we find the price of our bond? -P

5

5 100

• Strategy: Replicate the entire CF-stream we want to price • For there to be no arbitrage the price of the CF-stream must be the same as the price of the replication

How do we find the price of our bond? • Think of the bond as two zero-coupon bonds • Replicate each bond by depositing money in the bank, as before • Together, our two deposits will form a replicating portfolio • Note that the interest rate we get for a two year deposit may be different from that of a one year deposit – To indicate the maturity of an interest rate we typically use a time index: rt

How do we find the price of our bond • We want to replicate a cash flow of c at time T = 1 – Observe some interest rate, r1, that is valid over the time [0,1] – Today, deposit M1 such that M1(1+ r1) = c – Find that M1 = c/(1 + r1)

• We want to replicate a cash flow of FV + c at time T = 2 – Observe some interest rate, r2, that is valid over the time [0,2] – Today, deposit M2 such that M2(1+ r2)2 = FV + c – Find that M2 = (FV + c)/(1 + r2)2

• Our complete strategy costs M1 + M2 = c/(1 + r1) + (FV + c)/(1 + r2)2

Discounting • We say that P is the present value, PV, of the future cash flows • This process of calculating the PV of future CFs is called discounting • The market determines the appropriate interest rates, r1 and r2 • We are typically not explicit with the entire arbitrage argument

Discounting and prices • The price of a bond (or indeed any financial asset) is the sum of the present values of its future cash flows. • The price of a bond (or indeed any financial asset) is the sum of the present values of its future cash flows. That’s worth repeating.

Pricing formula and yield-to-maturity • When we have many CFs, discounting each one gets tedious • It would be useful with a compact pricing formula • To get one we will assume that interest rates are constant, i.e. rt = y for any t • We also have to introduce the concept of perpetuities

Perpetuities • A perpetuity is a never ending constant cash flow stream, e.g. an annual payment of c • How do we value such a thing? Set up a replicating portfolio • Let’s deposit some amount of money, M, in the bank and withdraw the interest every year • How large would M have to be in order to give an interest of c? My = c c M= y

What do we want to replicate? • One coupon stream (of c) from 1 to T • One large payment (of FV) at T • The present value of the FV is easy – PV(FV) = FV/(1+y)T

• The coupon stream, CS, can be viewed as the difference between two perpetuities: – One perpetuity starting at time 1, X1 – One perpetuity starting at time T+1, X2

• Its PV is the difference in the PV of X1 and X2 – PV(CS) = PV(X1) - PV(X2)

Replicating the coupon stream • The coupon stream, CS, can be viewed as the difference between two perpetuities: – One perpetuity starting at time 1, X1 – One perpetuity starting at time T+1, X2

• Its PV is the difference in the PV of X1 and X2 PV (CS ) = PV ( X 1 ) − PV ( X 2 ) PV (CS ) =

c c − y y

(1 + y )T

=

c 1  1 − y  (1 + y )T 

Pricing formula and yield-to-maturity • Adding the PV of the coupon stream and FV we get our pricing formula: P=

1 c 1 − y  (1 + y)T

 FV + T  (1 + y )

• Note that in practice interest rates are not constant • Instead we take P as given and define y as whatever interest rate satisfies the equation above. Expressed on an annual basis, we call this interest rate the yield-tomaturity (YTM). • Each bond has its own YTM

YTM and bond prices • This graph shows the price of a 30-year bond with a FV of $100 and a coupon rate of 10 % for different YTMs 350 300

Price

250 200 150 100 50 0 0%

5%

10%

15% YTM

20%

25%

YTM and bond prices • The bond price decreases with the YTM • The price is less sensitive to changes in the YTM when the YTM is high • When YTM = C = 10 %, P = FV = $100 – When P = FV (C = YTM), the bond trades at par – When P < FV (C < YTM), the bond trades at a discount – When P > FV (C > YTM), the bond trades at a premium

CGBs: Pricing conventions • Commonwealth Government Bonds (CGBs) are issued by the Australian government • Face value = $100 • Semi-annual coupons • Matures on the 15th of the maturity month/year

CGBs : Pricing conventions • Each semi-annual coupon is half the annual coupon. • It’s convenient to think of the yield on a semiannual basis, i.e. y = YTM/2. • Bond trades are settled after two working days

Pricing CGBs • To price a CGB which is settled on a coupon date, we simply add up the PV of the future cash flows as before:

c 1 P = 1 − y  (1 + y )T

 FV + T ( ) 1 y + 

Ex-interest bonds • CGB bonds are ex-interest if settled within 7 days before a coupon payment • We first calculate the value of the bond on the coupon date as above: P' =

1  FV c 1−  T  + y  (1 + y)  (1 + y)T

• Since this is a future cash flow, we must discount it to get the price today: P = PV(P’).

Ex-interest bonds • Suppose there are 184 days between coupons and the settlement date is 5 days before the next coupon. • There are 5/184 of a period until we get P’. Denote this fraction f. We must discount P’ f = 5/184 periods to get the present value, P:

P' P = PV ( P ' ) = (1 + y ) f

Cum-interest bonds • CGB bonds are cum-interest if settled with more than 7 days to the next coupon. In this case the coupon is included in the price. • Our total CF at the next coupon date is P’ + cpp • The present value of this is the price today: P '+ c P = PV ( P '+c ) = (1 + y ) f

CGB quoted prices • By convention, the market does not quote the settlement price P • If interest was continuous, we would get part of the coupon even if it’s settled ex-interest • The quoted or capital price (Padj) takes this into account by the following adjustments: Padj = P + c ⋅ f , if the bond is ex - interest Padj = P − c ⋅ (1 − f ), if the bond is cum - interest

Lecture 2 The term structure of interest rates

Learning outcomes • By the end of this lecture you should: – Be familiar with the concept of the term structure of interest rates and know what determines its shape – Be able to back out the term structure from a set of coupon bonds – Be able to profit from situations where different bonds imply different term structures – Be able to deduce forward rates from the term structure (and vice versa)

What is the term structure? • Suppose we can get a fixed interest rate for an investment starting today and ending at time t • We call this interest rate the and denote it rt (or 0rt to emphasize that it’s the t spot rate today, i.e. at time zero) – Note that the book denotes the short rates by – That’s not the standard notation and it blurs the distinction between yields and spot rates

• Together the spot rates make up the term structure of interest rates or the (pure) yield curve

Here it is

What determines the term structure? • Think of interest rates as (sort of) prices of future cash flows • Like other prices, they are set in equilibrium. When the prices • Typically (but not always) upwards sloping • We’ll discuss why the term structure looks like it does, and what implications it has

Is it useful? • The spot rate for a given maturity constitutes the alternative investment we used to set up the arbitrage trades to motivate discounting in lecture 1 • The spot rates are the appropriate discount rates for pricing (risk free) future cash flows • Take a three year bond with annual coupons: P0 =

(FV + c ) c c + + (1 + r1 ) (1 + r2 )2 (1 + r3 )3

Inferring the term structure • We directly observe market bond prices. These imply the term structure. • Zero coupon bonds have only one cash flow, so backing out the implied spot rate is straight forward: P=

FV

(1 + rt )t

1

 FV  t ⇔ rt =   −1  P 

• In practice, bonds with long maturities tend to pay coupons

Inferring the term structure • Consider the pricing equation of a two year bond at time 0:

( FV + c ) c P2 = + (1 + r1 ) (1 + r2 )2 • We have one equation and two unknowns • We have to back out the spot rates in an iterative manner. This method is called bootstrapping.

Example • First find r1 from a one-year zero coupon bond: P1 =

 FV FV ⇔ r1 =  (1 + r1 )  P1

 −1 

• Then substitute into the equation for P2;0. We now have one equation and one unknown: P2 =

( FV + c ) r c + ⇔ 2= (1 + r1 ) (1 + r2 )2

( FV + c )

 c  P −  2  − 1 r + 1 ( ) 1  

• We can repeat this for any number of steps

Recap: What is an arbitrage? • An arbitrage is a (set of) trades that generate zero cash flows in the future, but a positive and risk free cash flow today • A straight forward example is a violation of the law of one price • Our examples are based on the same principle but (slightly) more complex • They typically rely on constructing some synthetic instruments, i.e. portfolios that have the same CF consequences as the real instruments

Recap: What is an arbitrage? • We often refer to these synthetic instruments as replicating strategies or replicating portfolios • Once we have set up a replicating portfolio we can exploit any mispricings as if it was the real instrument • These trades, e.g. selling the real instrument and buying the synthetic instrument, are the arbitrage trades • Arbitrage traders will impose some structure on the term structure of interest rates

A numerical example • Suppose that the following bonds trade in the market: A -90.91

100

-79.72

0

100

-95.78

10

110

B

C

Get the arbitrage free price • Let’s back out the implied term structure from the pricing equations of bonds A and B: PA = 90.91 =

100 100 ⇔ r1 = −1 ≈ 10% 90.91 (1+ r1 )

PB = 79.72 =

100 100 ⇔ r2 = − 1 ≈ 12% 2 79.72 (1 + r2 )

• Now use the term structure to calculate the arbitrage free (or implied) price of bond C: PCImplied =

10 110 10 110 + = + ≈ 96.78 > 95.78 = PCObserved 2 2 (1 + r1 ) (1 + r2 ) 1.1 1.12

Set up the arbitrage trade • Bond C trades for less than the arbitrage free price, i.e. it is “too cheap” • There is an arbitrage opportunity • Strategy: Construct a synthetic version of Bond C from bonds A and B • Buy the underpriced real bond and sell the overpriced synthetic bond

Construct the synthetic bond • A synthetic bond replicates the CFs of the C bond so we want to achieve CF1S = CF1C = 10 CF2S = CF2C = 110

• Each A bond gives CF1 = 100 • The synthetic bond contains XA A-bonds, so that XA satisfies: X ACF1A = CF1C ⇔ X A100 = 10 ⇔ X A =

10 = 0.1 100

Construct the synthetic bond • Similarly, the synthetic bond contains XB Bbonds so that XB satisfies  X ACF1 A + X BCF1B = CF1C ⇔ 0.1⋅ 100 + X B ⋅ 0 = 10 110 X ⇒ = = 1. 1  B A B C 100  X ACF2 + X BCF2 = CF2 ⇔ 0.1 ⋅ 0 + X B ⋅100 = 110

• We have now constructed a synthetic C bond. • Sell this synthetic bond and buy the real bond.

Exploit the mispricing • Our CFs will be CF0 = (X A PA + X B PB ) − PC = ( 0.1⋅ 90.91+ 1.1⋅ 79.72) − 95.78 ≈ 96.78 − 95.78 = 1 CF1 = − X A CF1 A + CF1C = − 0.1⋅ 100 + 10 = 0 CF2 = − X B CF2B + CF2C = − 1.1⋅ 100 + 110 = 0

• We make a riskless profit at t = 0 • We can scale these trades up • When doing so, the supply and demand our trades create will push bond prices to their arbitrage free values

Reinvestment risk • Suppose we have an investment horizon of two years • If we have a coupon bond we must reinvest it at t = 1 to get all cash flows at t = 2 • Our cash flows at t = 2 will be: CF2 = FV + c + c(1 + 1r2) • Since 1r2 is unknown at t = 0, so is CF2

Holding period return • Our total CF at t = 2 will be: CF2 = FV + c + c(1 + 1r2) • Since this is risky, so is the return we make on this investment • To emphasize that we mean the (risky) investment return, we sometimes refer to this as the holding period return, HPR: HPR =
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